Assume that \(X\cong P\Omega _{n}(q)\)  with \(nq\)  is odd and \(n\geqslant 5\) . The group \(X_{x}\)  lies in a large maximal geometric
subgroup \(Y\)  of \(X\)  such that \(\left( \Phi _{(n-1)f}^{\ast }(p),\left|Y\right|\right) >1\)  by Theorem REF . Then \(Y\) 
is one of the groups listed in [1]}, Proposition 4.23, and hence \(Y\in \mathcal {C}_{1}(X)\) , since \(\left( \Phi _{(n-1)f}^{\ast }(p),\left|Y\right|\right) >1\) . Thus either \(X \cong P \Omega _{7}(q)\) , with \(q=3,5\) , and \(Y\)  of type \(O^{\varepsilon ^{\prime }}_{1}(q) \wr S_{7}\) , or \(Y \in \mathcal {C}_{1}(X)\) . The former is ruled out since \(13 \mid \left|X \right|\) , but \(13^{2} \nmid \left|X \right|\) , and \(13 \nmid \left|X_{x} \right|\) . Thus, \(Y \in \mathcal {C}_{1}(X)\)  and hence \(Y\cong \Omega _{n-1}^{-}(q).Z_{2}\)  is
the stabilizer of a non-singular point of \(PG_{n-1}(q)\)  by [2]},
Propositions 4.1.6.(II) and 4.1.20.(II). Moreover, \(X_{x}=Y\)  by Lemma REF (1) and by [2]}, Proposition 4.1.6.(I). Then \(k^{2}=q^{(n-1)/2}\frac{q^{(n-1)/2}-1}{2}\) , as \(k^{2}=\left[ X:X_{x}\right] \)  and \(q\)  is odd.
Hence,
\(k^{2}-1=\frac{(q^{(n-1)/2}+1)\left( q^{(n-1)/2}-2\right) }{2}\text{.}\) 
On the other hand, \(X\)  has subdegrees \(q^{(n-1)}-1\) , \(q^{(n-3)/2}\frac{(q^{(n-1)/2}+1)}{2}\)  and \((q-3)/2\)  times \(q^{(n-3)/2}(q^{(n-1)/2}+1)\)  (e.g.
see [4]}, p. 331). Thus \(q^{(n-3)/2}\frac{(q^{(n-1)/2}+1)}{2}\)  is a
subdegree for \(G\)  too, and hence \(k+1\mid \frac{(q^{(n-1)/2}+1)}{2}\)  by
Lemma REF (3). So, \(q^{(n-1)/2}-2\mid k-1\)  by (REF ), and hence \(q=2\) , \(n=5\)  and \(k^{2}=6\) , which is a contradiction.
